Question

1. Find the values of for which the gradient function of the curve = 23 + 32 −12 + 3 is zero.

Hence, find the equations of the tangents to the curve which are parallel to the −axis.

Answer 1

To find the values of x for which the gradient function of the curve is zero, we need to find the values of x that make the derivative of the curve equal to zero.

The given function is:
`\displaystyle\sf f(x)=23x^(3)+2x^(2)-12x+3`.

To find the derivative of the function, we differentiate each term with respect to x:

`\displaystyle\sf f'(x)=69x^(2)+4x-12`.

Now we set the derivative equal to zero and solve for x:

`\displaystyle\sf 69x^(2)+4x-12=0`.

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of x:

`\displaystyle\sf x=\frac{-b\pm \sqrt{b^(2)-4ac}}{2a}`.

Plugging in the values a=69, b=4, and c=-12, we get:

`\displaystyle\sf x=\frac{-4\pm \sqrt{(4)^(2)-4(69)(-12)}}{2(69)}`.

Simplifying further, we have:

`\displaystyle\sf x=(-4\pm √(16+3312))/(138)`.

`\displaystyle\sf x=(-4\pm √(3328))/(138)`.

`\displaystyle\sf x=(-4\pm 8√(13))/(138)`.

Therefore, the values of x for which the gradient function of the curve is zero are given by
`\displaystyle\sf x=(-4+8√(13))/(138)` and
`\displaystyle\sf x=(-4-8√(13))/(138)`.

To find the equations of the tangents to the curve that are parallel to the x-axis (horizontal tangents), we substitute these x-values into the original function
`\displaystyle\sf f(x)`. The resulting y-values will give the equations of the tangents.

For
`\displaystyle\sf x=(-4+8√(13))/(138)`:

`\displaystyle\sf y=f\left((-4+8√(13))/(138)\right)=23\left((-4+8√(13))/(138)\right)^(3)+2\left((-4+8√(13))/(138)\right)^(2)-12\left((-4+8√(13))/(138)\right)+3`.

For
`\displaystyle\sf x=(-4-8√(13))/(138)`:

`\displaystyle\sf y=f\left((-4-8√(13))/(138)\right)=23\left((-4-8√(13))/(138)\right)^(3)+2\left((-4-8√(13))/(138)\right)^(2)-12\left((-4-8√(13))/(138)\right)+3`.

These two equations represent the equations of the tangents to the curve that are parallel to the x-axis.